# Conoid

A Catalan surface all straight line generators of which intersect a fixed straight line, called an axis of the conoid. For example, a hyperbolic paraboloid is a conoid with two axes.

The position vector of a conoid is given by

$$r=\{u\cos v+\alpha f(v),u\sin v+\beta f(v),\gamma f(v)\},$$

where $\{\alpha,\beta,\gamma\}$ is a unit vector having the same direction as an axis of the conoid and $f(v)$ is some function. For a right conoid $\alpha=\beta=0$, $\gamma=1$, and then its axis is a line of striction. A right conoid with $f(v)=av$ is a helicoid.